Pythagoras

Considered a mathematician, but foremost a philosopher, Pythagoras was a very important figure in mathematics, astronomy, musical theory, and in the world's history. However, little in the way of reliable record is known about his life and accomplishments. The accounts of Pythagoras inventing the musical scale, performing miracles, and announcing prophecies are probably only legend, and appear to have little historical foundation. Scholars generally agree only upon the main events in his life, and usually combine together discoveries by Pythagoras with those by his band of loyal followers.

Pythagoras established in what is now the southeastern coast of Italy a philosophical, political, and religious society whose members believed that the world could be explained using mathematics as based upon whole numbers and their ratios. Their motto was "All is number." Even the words philosophy (or "love of wisdom") and mathematics (or "that which is learned") is believed to have been first used (and defined) by the Pythagoreans.

Many Pythagorean beliefs (such as secrecy, vegetarianism, periods of food abstinence and silence, refusal to eat beans, refusal to wear animal skins, celibacy, self-examination, immortality, and reincarnation) were directed as "rules of life." The main focus of Pythagorean thought was ethics, developed primarily within philosophy, mathematics, music, and gymnastics. The beliefs of the society were that reality is mathematical; philosophy is used for spiritual purification; the soul is divine; and certain symbols possess mystical significance. Both men and women were permitted to become members. In fact, several female Pythagoreans became noted philosophers.*

*Aesara of Lucania was a Pythagorean philosopher known for her theory of the tripart soul, which she believed consisted of the mind, spiritedness, and desire.

How Pythagoreans Conceptualized Numbers

Pythagoreans believed that all relationships could be reduced to numbers in order to account for geometrical properties. This generalization originated from the observation that whenever the ratios of lengths of strings were whole numbers, harmonious tones were produced when these strings were vibrated.

The society studied properties of numbers that are familiar to modern mathematicians, such as even and odd numbers, prime and square numbers. From this viewpoint, the Pythagoreans developed the concept of number, which became their dominant principle of all proportion, order, and harmony in the universe.

The society also believed in such numerical properties as masculine or feminine, perfect or incomplete, and beautiful or ugly. These opposites, they believed, were found everywhere in nature, and the combination of them brought about the harmony of the world.

The primary belief of Pythagoreans in the sole existence of whole numbers was later challenged by their own findings, which proved the existence of "incommensurables," known today as irrational numbers . What is commonly called the "first crisis in mathematics" caused a scandal within the society, so serious that some members tried to suppress the knowledge of the existence of incommensurables.

The Pythagorean philosophy was dominated by the ideal that numbers were not only symbols of reality, but also were the final substance of real things, known as "number mysticism." They held, for example, that one is the point, two the line, three the surface, and four the solid. Seven was considered the destiny that dominates human life because infancy ends there, and also because the number was associated with the seven wandering stars. Moreover, Pythagoreans believed that maturity began at age 14, marriage occurred in the twenty-first year, and 70 years was the normal life span. Ten was identified as the "perfect number" because it was the sum of one, two, three, and four.

Pythagorean Contributions to Mathematics

The formalization of mathematics with the use of axiomatic systems was the most profound contribution that the Pythagorean society made to mathematics. Pythagoreans developed this significant concept by showing that arbitrary laws of empirical geometry could be proved as logical conclusions from a small number of axioms, or postulates. Typical of the developed axioms was "A straight line is the shortest distance between two points."

From these axioms, a number of theorems about the properties of points, lines, angles, curves, and planes could be logically deduced. These theorems include the famous Pythagorean theorem, which states that "the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides." Another theorem states that the sum of the interior angles of any triangle is equal to the sum of two right angles.

The Pythagorean Theorem

The Pythagoreans knew that any triangle whose sides were in the ratio 3:4:5 was a right-angled triangle. Their desire to find the mathematical harmonies of all things led them to prove the geometric theorem, today named for Pythagoras. The earlier Egyptians stated this theorem as an empirical relationship and, as far as is known today, the Pythagoreans were the first to prove it.

The Pythagorean (hypotenuse) theorem states that the square of the hypotenuse of a right-angle triangle (c ) is equal to the sum of the squares of the other two sides (a and b ), shown as c2 = a2 + b2. The numbers 3, 4, and 5 are called Pythagorean numbers (52 = 32 + 42, or 25 = 9 + 16). However, the Pythagoreans did not consider the square on the hypotenuse to be that number (c ) multiplied by itself (c2). Instead, it was conceptualized as a geometrical square (C ) constructed on the side of the hypotenuse, and that the sum of the areas of the two squares (A and B ) is equal to the area of the third square (C ), as shown below.

Astronomy and the Pythagoreans

In astronomy, the Pythagoreans produced important advances in ancient scientific thought. They were the first to consider the Earth as a sphere revolving with the other planets and the Sun around a universal "central fire." Ten planets were believed to exist in order to produce the "magical" number of 10. This arrangement was explained as the harmonious arrangement of bodies in a complete sphere of reality based on a numerical pattern, calling it a "harmony of sphere." The Pythagoreans also recognized that the orbit of the Moon was inclined to the equator of the Earth, and were one of the first to accept that Venus was both the evening star and the morning star.

Even though much of the Pythagorean doctrine consisted of numerology and number mysticism, their influence in developing the idea that nature could be understood through mathematics and science was extremely important for studying and understanding the world in which we live.

Philip, James A. Pythagoras and Early Pythagoreanism. Toronto: University of Toronto Press, 1966.

MAGIC OVER MATHEMATICS

During the time of Pythagoras, most people either believed that the world could only be explained by magic or that it could not be explained at all. Thus, many people did not attempt to understand mathematics.

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Pythagoras

Encyclopedia of World Biography
COPYRIGHT 2004 The Gale Group Inc.

Pythagoras

The Greek philosopher, scientist, and religious teacher Pythagoras (ca. 575-ca. 495 B.C.) evolved a school of thought that accepted the transmigration of souls and established number as the principle in the universe.

Born on the island of Samos, Pythagoras was the son of Mnesarchus. He fled to southern Italy to escape the tyranny of Polycrates, who came to power about 538, and he is said to have traveled to Egypt and Babylon. He and his followers became politically powerful in Croton in southern Italy, where Pythagoras had established a school for his newly formed sect. It is probable that the Pythagoreans took positions in the local government in order to lead men to the pure life which their teachings set forth. Eventually, however, a rival faction launched an attack on the Pythagoreans at a gathering of the sect, and the group was almost completely annihilated. Pythagoras either had been banished from Croton or had left voluntarily shortly before this attack. He died in Metapontum early in the 5th century.

Religious Teachings

Pythagoras and his followers were important for their contributions to both religion and science. His religious teachings were based on the doctrine of metempsychosis, which held that the soul was immortal and was destined to a cycle of rebirths until it could liberate itself from the cycle through the purity of its life. A number of precepts were drawn up as inviolable rules by which initiates must live.

Pythagoreanism differed from the other philosophical systems of its time in being not merely an intellectual search for truth but a whole way of life which would lead to salvation. In this respect it had more in common with the mystery religions than with philosophy. Several taboos and mystical beliefs were taught which sprang from a variety of primitive sources such as folk taboo, ritual, and sympathetic magic and were examples of the traditional beliefs that the Greeks continued to hold while developing highly imaginative and rational scientific systems.

An important underlying tenet of Pythagoreanism was the kinship of all life. A universal life spirit was thought to be present in animal and vegetable life, although there is no evidence to show that Pythagoras believed that the soul could be born in the form of a plant. It could be born, however, in the body of an animal, and Pythagoras claimed to have heard the voice of a dead friend in the howl of a dog being beaten.

The number of lives which the soul had to live before being liberated from the cycle is uncertain. Its liberation came through an ascetic life of high moral and ethical standards and strict adherence to the teachings and practices of the sect. Pythagoras himself claimed to remember four different lives. Followers of the sect were enjoined to secrecy, although the discussions of Pythagoras's teachings in other writers proved that the injunction was not faithfully observed.

Mathematical Teachings

The Pythagoreans posited the dualism between Limited and Unlimited. It was probably Pythagoras himself who declared that number was the principle in the universe, limiting and giving shape to matter. His study of musical intervals, leading to the discovery that the chief intervals can be expressed in numerical ratios between the first four integers, also led to the theory that the number 10, the sum of the first four integers, embraced the whole nature of number.

So great was the Pythagoreans' veneration for the "Tetractys of the Decad" (the sum of 1 + 2 + 3 + 4) that they swore their oaths by it rather than by the gods, as was conventional. Pythagoras may have discovered the theorem
which still bears his name (in right triangles, the square on the hypotenuse equals the sum of the squares on the other sides), although this proposition has been discovered on a tablet dating from the time of the Babylonian king Hammurabi. Regardless of their sources, the Pythagoreans did important work in systematizing and extending the body of mathematical knowledge.

As a more general scheme, the Pythagoreans posited the two contraries, Limited and Unlimited, as ultimate principles. Numerical oddness and evenness are equated with Limited and Unlimited, as are one and plurality, right and left, male and female, motionlessness and movement, straight and crooked, light and darkness, good and bad, and square and oblong. It is not clear whether an ultimate One, or Monad, was posited as the cause of the two categories.

Cosmological Views

As a result of their religious beliefs and their careful study of mathematics, the Pythagoreans developed a cosmology which differed in some important respects from the world views of their contemporaries, the most important of which was their view of the earth as a sphere which circled the center of the universe. The center of this system was fire, which was invisible to man because his side of the earth was turned from it. The sun reflected that fire; there was a counterearth closer to the center, and the other five planets were farther away and followed longer courses around the center. It is not known how much of this theory was attributable to Pythagoras himself. Later writers ascribe much of it to Philolaos (active 400 B.C.), although it circulated as a view of the school as a whole.

The systematization of mathematical knowledge carried out by Pythagoras and his followers would have sufficed to make him an important figure in the history of Western thought. However, his religious sect and the asceticism which he taught, embracing as it did a vast number of ancient beliefs, make him one of the great teachers of religion in the ancient Greek world.

Further Reading

Pythagoras left no written works. A first-rate technical book, J. A. Philip, Pythagoras and Early Pythagoreanism (1966), separates the valid from the spurious among the legends that surround Pythagoras and his views. An excellent and thorough treatment of the evidence for his life and teachings is in W. K. C. Guthrie, A History of Greek Philosophy (3 vols., 1962-1969). A good account of Pythagoras and his followers is in Kathleen Freeman, The Pre-Socratic Philosophers (1946; 3d ed. 1953), and G. S. Kirk and J. E. Raven, The Presocratic Philosophers (1962). Briefer treatments of the Pythagoreans and the intellectual currents of their time are in the standard histories of Greek literature, such as Albin Lesky, A History of Greek Literature (trans. 1966), or in accounts of Greek philosophy, such as John Burnet, Greek Philosophy (1914). □

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Pythagoras

The Greek philosopher, scientist, and religious teacher Pythagoras developed a school of thought that accepted the passage of the soul into another body and established many influential mathematical and philosophical theories.

Early life

Born on the island of Samos, off Greece, in the Mediterranean Sea, Pythagoras was the son of Mnesarchus. Little is known about his early life. After studying in Greece, he fled to southern Italy to escape the harsh rule of Polycrates (died c. 522 b.c.e.), who came to power about 538 b.c.e. Pythagoras is said to have traveled to Egypt and Babylon during this time.

Pythagoras and his followers became politically powerful in Croton in southern Italy, where Pythagoras had established a school for his newly formed sect, or group of followers. It is probable that the Pythagoreans
took positions in the local government in order to lead men to the pure life that was directed by their teachings. Eventually, however, a rival group launched an attack on the Pythagoreans at a gathering of the sect, and the group was almost completely destroyed. Pythagoras either had been forced to leave Croton or had left voluntarily shortly before this attack. He died in Metapontum early in the fifth century b.c.e.

Religious teachings

Pythagoras and his followers were important for their contributions to both religion and science. His religious teachings were based on the doctrine (teaching) of metempsychosis, which teaches that the soul never dies and is destined to a cycle of rebirths until it is able to free itself from the cycle through the purity of its life.

Pythagoreanism differed from the other philosophical systems of its time in being not merely an intellectual search for truth but a whole way of life which would lead to salvation, or to be delivered from sin. An important part of Pythagoreanism was the relationship of all life. A universal life spirit was thought to be present in animal and vegetable life, although there is no evidence to show that Pythagoras believed that the soul could be born in the form of a plant. It could be born, however, in the body of an animal, and Pythagoras claimed to have heard the voice of a dead friend in the howl of a dog being beaten.

Mathematical teachings

The Pythagoreans presented as fact the dualism (that life is controlled by opposite forces) between Limited and Unlimited. It
was probably Pythagoras who declared that numbers could uncover the secrets of the universe, limiting and giving shape to matter (anything that takes up space). His study of musical intervals, leading to the discovery that the chief intervals can be expressed in numerical ratios (relationships between numbers) between the first four integers (positive whole numbers), also led to the theory that the number ten, the sum of the first four integers, embraced the whole nature of number.

So great was the Pythagoreans' respect for the "Tetractys of the Decad" (the sum of 1 + 2
+ 3 + 4) that they swore their oaths (promises) by it rather than by the gods, as was normal during his day. Pythagoras may have discovered the theorem which still bears his name (in right triangles [triangle with one angle equal to 90 degrees], the square on the hypotenuse equals the sum of the squares on the other sides), although this proposal has been discovered on a writing stone dating from the time of the Babylonian king Hammurabi (died c. 1750 b.c.e.). Regardless of their sources, the Pythagoreans did important work in extending the body of mathematical knowledge.

As a more general outline, the Pythagoreans presented the two contraries (opposites), Limited and Unlimited, as ultimate principles, or truths. Numerical oddness and evenness are equated with Limited and Unlimited, as are one and plurality (many), right and left, male and female, motionlessness and movement, straight and crooked, light and darkness, and good and bad. It is not clear whether an ultimate One, or Monad, was presented as the cause of the two categories.

Cosmological views

The Pythagoreans, as a result of their religious beliefs and careful study of mathematics, developed a cosmology (dealing with the structures of the universe) which differed in some important respects from the world views at the time, the most important of which was their view of the Earth as a sphere which circled the center of the universe. It is not known how much of this theory was credited to Pythagoras himself.

The mathematical knowledge carried out by Pythagoras and his followers would have been enough to make him an important figure in the history of Western thought. However, his religious sect and the self-discipline and dedication which he taught, embracing as it did a vast number of ancient beliefs, make him one of the great teachers of religion in the ancient Greek world.

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Pythagoras

PYTHAGORAS.Pythagoras (c. 580–c. 580 B.C.E.) was a Greek mathematician, philosopher, and mystic. He wrote nothing himself, so his ideas survive through the writings of others, including Aristotle. Many people are familiar with him as the mathematician who formulated the Pythagorean theorem in geometry that relates the lengths of the sides in a right triangle. Others know him as a mystic and the first person known to be motivated by moral and philosophical concerns to adopt a vegetarian diet.

The schools and societies Pythagoras founded in the southern Italian area of Magna Graecia flourished for a while, and they developed and spread many of his concepts, which were later adopted and expanded by others. These concepts include bodily humors (evident in modern descriptions of melancholic and phlegmatic personalities), a tripartite soul, reincarnation, and the numerical ratios that determine the concordant intervals of the musical scales. Permeating all of his thoughts was the idea that all things are numbers. Numbers (individuals, groups, and series) were imbued with mystical properties that were carefully guarded and only shared among initiates to the Pythagorean schools founded by him or his disciples.

Pythagoras and his followers practiced one of the first recorded diets known as vegetarianism. He advocated a diet devoid of the flesh of slaughtered animals partially because he felt food influenced the distribution of the bodily humors and thereby the health of the individual and partially because it would prevent the killing of a reincarnated individual and its transmigrated soul. Up until the late nineteenth century non–meat eaters were generally known as "Pythagoreans."

Pythagoras is also alleged to have admonished his disciples to abstain from eating beans. Ancient and medieval writers ingeniously ascribed this pronouncement to the belief that beans contained or transmitted souls. The Greek phrase supporting this gastronomic recommendation, however, could also be construed to imply that his followers should avoid politics. Black and white beans were used as counters in voting in Magna Graecia. The school Pythagoras founded there became actively involved in the populist political views that gained ascendancy in the town of Kroton, where he lived for many years. Later an opposing aristocratic party gained control of the city and banished him and his followers for their political views and activism. Pythagoras died in exile. His supposed warning to "abstain from beans" is therefore thought to have meant "avoid politics." Alternatively he may have realized that eating undercooked broad (fave) beans (Vica faba vulgaris), a common food of the Mediterranean region, produced a severe hemolytic anemia (favism) in some people. Interestingly the same mutant gene that makes people sensitive to favism also increases their resistance to the malarial parasite, possibly accounting for the widespread presence of the mutant gene in regions with endemic malaria.

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Pythagoras

The Columbia Encyclopedia, 6th ed.

Copyright The Columbia University Press

Pythagoras (pĬthăg´ərəs), c.582–c.507 BC, pre-Socratic Greek philosopher, founder of the Pythagorean school. He migrated from his native Samos to Crotona and established a secret religious society or order similar to, and possibly influenced by, the earlier Orphic cult. We know little of his life and nothing of his writings. Since his disciples came to worship him as a demigod and to attribute all the doctrines of their order to its founder, it is virtually impossible to distinguish his teachings from those of his followers. The Pythagoreans are best known for two teachings: the transmigration of souls and the theory that numbers constitute the true nature of things. The believers performed purification rites and followed moral, ascetic, and dietary rules to enable their souls to achieve a higher rank in their subsequent lives and thus eventually be liberated from the
"wheel of birth."
This belief also led them to regard the sexes as equal, to treat slaves humanely, and to respect animals. The highest purification was
"philosophy,"
and tradition credits Pythagoras with the first use of the term. Beginning with the discovery that the relationship between musical notes could be expressed in numerical ratios (see Greek music), the Pythagoreans elaborated a theory of numbers, the exact meaning of which is still disputed by scholars. Briefly, they taught that all things were numbers, meaning that the essence of things was number, and that all relationships—even abstract ethical concepts like justice—could be expressed numerically. They held that numbers set a limit to the unlimited—thus foreshadowing the distinction between form and matter that plays a key role in all later philosophy. The Pythagoreans were influential mathematicians and geometricians, and the theorem that bears their name is witness to their influence on the initial part of Euclidian geometry. They made important contributions to medicine and astronomy and were among the first to teach that the earth was a spherical planet, revolving about a fixed point. At the end of the 5th cent. BC the Pythagoreans were forced to flee Magna Graecia when people grew enraged at their interference with traditional religious customs; many were killed. A short-lived Neo-Pythagoreanism developed at the beginning of the Christian era; it borrowed some elements from Jewish and Hellenistic thought and greatly emphasized the mystical element in Pythagorean ideas.

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Pythagoras

Pythagoras (c.580–500 bc), Greek philosopher; known as Pythagoras of Samos. Pythagoras sought to interpret the entire physical world in terms of numbers, and founded their systematic and mystical study; he is best known for the theorem of the right-angled triangle. His analysis of the courses of the sun, moon, and stars into circular motions was not set aside until the 17th century.

Pythagoras also founded a secret religious, political, and scientific sect in Italy: the Pythagoreans held that the soul is condemned to a cycle of reincarnation, from which it may escape by attaining a state of purity.Pythagoras' theorem the theorem attributed to Pythagoras that the square on the hypotenuse of a right-angled triangle is equal in area to the sum of the squares on the other two sides.Pythagorean letter the Greek letter Y, used by Pythagoras as a symbol of the two divergent paths of virtue and of vice.Pythagorean system the system of astronomy proposed by Pythagoras, in which all celestial bodies, including the earth, were held to revolve around a central fire (not the sun, but presumably identified with the sun, resulting in the system being assumed identical with the Copernican system).

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Pythagoras of Rhegium

The Columbia Encyclopedia, 6th ed.

Copyright The Columbia University Press

Pythagoras of Rhegium (pĬthăg´ərəs rē´jəm), fl. 5th cent. BC, Greek sculptor. In a signature on a pedestal at Olympia he declares himself a Samian, but the period of his training and work belongs to Rhegium, Italy. As no works are known that can with certainty be identified as his, his fame depends upon the statements of those who saw his statues and named them. They were mainly of athletes and mark a step in the transition between the archaic and the classical styles. He is said to have been the first to represent hair, veins, and muscles naturally and the first to aim at rhythm and symmetry in sculpture. Among his statues were a portrait of the boxer Euthymus, a figure of a man singing to a lyre, and one of Apollo shooting the Python with his arrows.

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Pythagoras

Pythagoras (c.580–500 bc) Greek philosopher and founder of the Pythagorean school. The Pythagoreans were bound to their teacher by rigid vows and were ascetic in their way of life. They believed in the transmigration of souls and that numbers and their interrelationships constitute the true nature of the universe. Pythagoras is credited with advances in mathematics and geometry, medicine and philosophy. The famous theorem, that the square of the hypotenuse of a right-angled triangle equals the sum of the squares of the other two sides, is named after him, but was already known to the Egyptians and Babylonians. His school was suppressed at the end of the 6th century, but the Romans revived its doctrines c.500 years later.

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